2=1

Can you point out the flaw in the supposed proof that 1 = 0.9999... ?

Simonm: From a common sense or intuitive point of view, there is nothing wrong with the proof that recurring .99999 equals one. Almost every serious mathematician up to the end of the 19th and many in the early 20th century would have accepted it as valid. For all practical purposes it is valid. After all, there is a valid proof that recurring .99999 has one for a limit. How far off base could the proof's conclusion be?

A serious mathematician would not claim that the proof is erroneous. He would claim that it is invalid due to undefined operations. In particular, the multiplication of recurring .9999 by ten and the subsequent subtraction of recurring .9999 from the product. These steps require doing operations on decimal fractions of unbounded length, which are undefined operations.

For any finite number of recurring nines, the proof breaks down. As some posters have already mentioned, perhaps then is a nine missing or unaccounted for in the proof. For a finite number of nines, you merely prove that recurring nines equals recurring nines. It requires a leap of faith to assume that for infinitely many nines the proof would work.

Some time starting in the late 19th or early 20th century, mathematicians became very picky about what constituted a valid proof and what are acceptable operations. The new attitude came about because some extremely subtle errors crept into the mathematical discipline. Proofs which later turned out to be invalid (and contradictory to other proofs) were discovered.

The problems were attributed to the use of common notions rather than formal logic.

Concepts relating to infinity are primarily (perhaps exclusively) dealt with in Set Theory. In analysis, calculus, algebra, analytical geometry, et cetera only finite operations are defined. Terminology like increases without bound is used to avoid words like infinity. When dealing with limits, a concept like the following is used.

• Consider the difference between the sum of a series and its proposed limit.
• Choose any small finite amount which we will call Delta.
• If for any arbitrarily small value of Delta a numbers of terms can be specified which makes the difference smaller than Delta, then the series sum is said to have or to approach the proposed limit.

The above definition does not say anything about an infinite number of terms. It does not refer to some value becoming infinite. It does not require any operations involving an infinite number of terms. It merely says "You choose a small finite Delta and I will tell you a finite numbers of terms that will result in the difference between the sum and the limit being less than Delta."

guys, that whole thing with .9999... is only true because you are treating infinity as a number. Infinity is not a number, rather, it is a mathematical animal in and of itself that describes an action. Of course regular mathematic breaks down when you stop using actual numbers. Dammit. I'm turning into math geek. i'm leaving before i start going to geek parties and asking for some Pi.

1 - .9999~ = 0 is flawed logically. With an infinite set of 9's there would never be a "Last" 9 to borrow from.